Fundamental Physical Constants 1986 Adjustments

E. Richard Cohen, Thousand Oaks CA and Barry N. Taylor, Gaithersburg (Rockwell International Science Center and National Bureau of Standards)

Over the past twenty years a surpri­ Table 1. 1986 Adjustment of the fundamental constants. singly strong link has developed bet­ Comparison of the initial and final stages of the multivariate analysis. ween the applied science of metrology and atomic, molecular and solid state Initial Final physics. The techniques of frequency (least-squares, (extended multiplication, and the precise measure­ external error) least-squares) ment of frequency at infrared and visible N 35 22 wavelengths, have reached such a level V 30 17 of development that the metre has been x 2 106.6 17.01 1.89 1.00 redefined in terms of the distance travell­ R b ed by light in a given time 1, 2). The direct α -1 137.035 996(11) 137.035 9895(61) linking of atomic lattice spacings to opti­ Kv 1 -7.24(54) x 10- 6 1 - 7.59(30)x10-6 cal wavelengths has produced a signifi­ KΩ 1 - 1.524(92) x 10-6 1 - 1.563(50)x10-6 cant improvement in the determination d220 192.015553(74) pm 192.015 540(40) pm of the Avogadro constant 3, 4). Impres­ µµ/µP 3.183 345 71(87) 3.183 345 47(47) sive progress has been made in the pre­ cision of measurement of the electron weighted mean of the observational 18). These algorithms use the consisten­ anomalous magnetic moment 5) as well equations that gives the smallest statis­ cy of the data to provide additional infor­ as the numerical evaluation of the asso­ tical variances consistent with the con­ mation with which to improve these a ciated sixth- and eighth-order Feynman straints. priori estimates of the variances. diagrams 6). The most striking metrolo­ The weight ωi associated with each gical advance occurred when K. von experimental datum is the reciprocal of Input Data Klitzing in 1980 observed the quantiza­ the variance, or statistical mean square As in previous adjustments, the data tion of electrical conductance 7) and error, σ2i which is however only available are divided into two categories : achieved not only a direct macroscopic as an a priori estimate s2i and is itself a) the more precise data (auxiliary con­ measurement of the fine-structure con­ uncertain. The usual least-squares pro­ stants) that are not subject to adjust­ stant, but the 1985 Nobel Prize in Phy­ cedure multiplies the calculated uncer­ ment because of their relatively low sics as well 8). tainties of an adjustment by the Birge uncertainties, and The CODATA Task Group on Funda­ ratio, RB = (x2/v)1/2, to rescale the b) the less precise or stochastic data mental Constants has recently comple­ weights and give a value of x2 equal to that are subject to adjustment. ted a new evaluation of the fundamental its expectation value v. This is equiva­ There is no formal basis for separation physical constants 9) taking into ac­ lent to an a posteriori evaluation of the into these two categories except that a count the truly extraordinary amount of 'error associated with unit weight' and is variable with an uncertainty much experimental and theoretical work that valid if the assigned uncertainties have smaller than that of other variables to has become available since the previous only relative significance, or if the syste­ which it is connected will be only slight­ adjustment in 1973 10). matic errors of all input data are roughly ly altered by the adjustment and can Because of past problems associated similar. However, when the data come hence be treated as a constant. The with the statistical treatment of such a from different and unrelated sources uncertainty of an auxiliary constant is diverse set of experimental and theoreti­ with broadly different physical content, typically one twentieth the uncertainty cal data, increased attention was direc­ a uniform expansion of all uncertainties of the stochastic datum with which it ted in the 1986 analysis to questions of can hardly be justified. In such a case, appears. All the auxiliary constants have statistical validity. The least-squares ap­ any rescaling of the assigned weights uncertainties not greater than 0.02 ppm proach to the analysis of such data has should consider all other information while the 38 items of stochastic data been described in detail in previous re­ that may be available concerning the have uncertainties in the range 0.05 views 11, 12, 13, but in brief, each experi­ uncertainty assignment of each indivi­ ppm to 10 ppm mental result with its estimated uncer­ dual datum. In analyzing the 1986 input On the basis of a preliminary screen­ tainty represents a constraint on the data we have considered not only the ing, three of the 38 items were identified values of a set of physical quantities, ex­ usual least-squares algorithm, but also as inappropriate for further considera­ pressed as an algebraic relationship in­ the algorithm proposed by Tuninskii and tion in the adjustment: one because it volving the auxiliary constants, the Kholin in 1975 14) of the Mendeleyev In­ represented an uncompleted measure­ unknown stochastic data and their un­ stitute of Metrology in Leningrad, as well ment; a second because it had recently known experimental and theoretical as a modification of it suggested by been shown to be in error; and the third errors. The least-squares solution de­ Taylor in 1982 15), and the extended because it required for its interpretation termines the "best" values of these least-squares algorithms described by an inadequately developed theoretical unknown quantities by finding the Cohen in 1976 16), 1978 17) and 1980 expression. The remaining 35 items for- 65 med the basis for an analysis with five Table 2. Summary of the 1986 recommended values of the fundamental physical unknowns. These unknowns were the constants. inverse fine-structure constant a-1, the A list of some fundamental constants of physics and chemistry based on a least-squares ratio of the standard volt (defined and adjustment with 17 degrees of freedom. The digits in parentheses are the one-standard- maintained by the Josephson effect) to deviation uncertainty in the last digits of the given value. Since the uncertainties of many of these entries are correlated, the full covariance matrix must be used in evaluating the the SI volt Kv, the ratio of the standard uncertainties of quantities computed from them. ohm (maintained in terms of a set of standard resistance coils, corrected for drift to the value on 1 Jan, 1985) to the Relative SI ohm KΩ, the lattice spacing of the Quantity Symbol Value Units uncertainty silicon lattice (at 22.5°C in vacuum) (ppm) d220, and the ratio of the muon magne­ tic moment to the proton magnetic mo­ speed of light in vacuum c 299 792 458 m s-1 (exact) ment, µµ/µP. The multivariate analysis permeability of vacuum µ0 4δπx10-7 N A- 2 was carried out using the several algo­ =12.566 370 614... 10-7 N A- 2 (exact) rithms mentioned above ; the details are permittivity of vacuum ϵ0 1/µ0c² (exact) given in CODATA Bulletin No. 63 9). =8.854187 817... 10-12 F m- 1 Newtonian constant of gravitation G 6.672 59(85) 10-11 m3 kg-1 s- 2 128 From the results of applying the Planck constant h 6.626 0755(40) 10-34 J s 0.60 various algorithms to the data and from h/2π h 1.054 572 66(63) 10- 34J s 0.60 a consideration of the effects on con­ elementary charge e 1.602 177 33(49) 10- 19C 0.30 sistency produced by eliminating data, magnetic flux quantum, h/2e Φ0 2.067 834 61(61) 10-15 Wb 0.30 we deleted 11 observations of very low Josephson frequency-voltage ratio 2e/h 4.835 9767(14) 1014 Hz V- 1 0.30 weight and two other older observations quantized Hall resistance, RH 25 812.8056(12) 0.045 which were of low weight and some­ h/e2 = µ0c/2a electron mass me 9.109 3897(54) 10-31 kg 0.59 what discrepant relative to their claimed 5.485 799 03(13) 10-4 u 0.023 precision. As a result, 22 items of sto­ in electron volts, mec2/{e} 0.510 999 06(15) MeV 0.30 chastic data remained to define the data electron specific charge — e/m e -1.758 819 62(53) 1011 C kg-1 0.30 set for the 1986 recommended values of muon mass mµ 1.883 5327(11) 10-28 kg 0.61 the fundamental physical constants. 0.113428 913(17) u 0.15 MeV 0.32 This final solution is compared with the in electron volts, mµc2/{e} 105.658 389(34) muon-electron mass ratio mµ/me 206.768 262(30) 0.15 initial adjustment with 35 observations proton mass mP 1.672 6231(10) 10- 27kg 0.59 in Table 1. The uncertainties given for the 1.007 276 470(12) u 0.012 initial adjustment (with X2 = 106.6 and in electron volts, mpc2/{e} 938.272 31(28) MeV 0.30 RB = 1.89) are computed from external proton-electron mass ratio mp/me 1836.152 701(37) 0.020 consistency. These uncertainties, which neutron mass mn 1.674 9286(10) 10- 27kg 0.59 are expanded by the factor RB from the 1.008 664 904(14) u 0.014 939.565 63(28) MeV 0.30 internally computed values, are still in electron volts, mnc2/{e} neutron-electron mass ratio mn/me 1838.683 662(40) 0.022 significantly smaller than the correspon­ neutron-proton mass ratio mn/m P 1.001 378 404(9) 0.009 ding uncertainties of the 1973 adjust­ α 10- 3 0.045 ment. Even though this solution con­ fine-structure constant, µ0ce2/2/t 7.297 353 08(33) inverse fine-structure constant α -1 137.035 9895(61) 0.045 tains some discordant data, none of Rydberg constant, mecα2/2h R∞ 10 973 731.534(13) m-1 0.0012 these quantities differs from its final Bohr radius, α/4πR∞ a0 0.529 177 249(24) 10-10 m 0.045 recommended value by more than 0.7 Compton wavelength, h/mec Ac 2.426 310 58(22) 10-12 m 0.089 standard deviations of that difference, δλc/2π = αa0 = α2 / 4πR∞ λc 3.86 159 323(35) 10-13 m 0.089 and only Kv differs from the final recom­ classical electron radius, α 2a0 re 2.817 940 92(38) 10-15 m 0.13 10-28 m2 0.27 mended value by more than one stan­ Thomson cross section, (8π/3 )re2 0.665 246 16(18) 0.34 dard deviation of that value. Bohr magneton, eh/2me µB 927.401 54(31) 10- 26 J T - 1 nuclear magneton, eh/2m p µN 0.505 078 66(17) 10-26 J T - 1 0.34 Table 2 gives a selected list of values electron magnetic moment µe 928.477 01(31) 10-26 J T - 1 0.34 of the fundamental constants of physics in Bohr magnetons µe/µB 1.001 159 652193(10) 1x 10-5 and chemistry ; Table 3 presents a set of in nuclear magnetons µe/µN 1838.282 000(37) 0.020 related values, such as the quantities proton magnetic moment µP 1.410 607 61(47) 10-26 J T - 1 0.34 VBI-76, ΩBI85 and d220 that are a neces­ in Bohr magnetons Pp/PB 1.521 032 202(15) 10-3 0.010 sary part of the data of the adjustment, in nuclear magnetons Pp/ pn 2.792 847 386(63) 0.023 diamagnetic shielding correction but cannot be considered as fundamen­ for protons in pure water, tal in the same sense as the quantities of spherical sample, 25 °C, 1 — µ'P/µP> σH2O 25.689(15) 10-6 - Table 2. shielded proton moment 1.410 571 38(47) 10-26 J T - 1 0.34 (H2O, sph., 25 °C) Comparisons and Discussion in Bohr magnetons µ'p/PB 1.520 993 129(17) 10-3 0.011 The precision of the 1986 recom­ in nuclear magnetons µ'p/µN 2.792 775 642(64) 0.023 mended values is roughly an order of proton gyromagnetic ratio γp 26 752.2128(81) 104 s- 1 T - 1 0.30 magnitude better than that of their 1973 γP/2 π 42.577 469(13) MHz T - 1 0.30 counterparts ; the precision of the uncorrected (H2O, sph., 25 °C) γp 26 751.5255(81) 104 s- 1 T - 1 0.30 Rydberg constant R∞ is improved by a γ'p/2π 42.576 375(13) MHz T - 1 0.30 muon-proton factor of 60 while that of mp/me and a, magnetic moment ratio µµ/ µP 3.183 345 47(47) 0.15 by factors of approximately 20. The neutron magnetic moment µn 0.966 237 07(40) 10-26 J T - 1 0.41 most significant revision is the change in in Bohr magnetons µn/µB 0.001 041 875 63(25) 0.24 KV and the resultant 7.75 ppm change in nuclear magnetons µn/µN 1.913 042 75(45) 0.24

66 Table 2. Summary of the 1986 recommended values of the fundamental physical in the recommended value of 2e/h: the constants (continued). 1986 value is higher than the 1973 value by three times the standard deviation of Relative the latter. The Josephson frequency-vol­ Quantity Symbol Value Units uncertainty tage ratio (483 594.0 GHz/V) adopted (ppm) by the Consultative Committee on Elec­ tricity in 1972, which was intended to neutron-electron reproduce the SI value and which forms magnetic moment ratio µn/µ e 0.001040 668 82(25) 0.24 neutron-proton the basis of the legal representation of magnetic moment ratio µn/µp 0.684 979 34(16) 0.24 the volt in many countries, is too small by about 8 ppm. This unsatisfactory Avogadro constant Na ,L 6.022 1367(36) 1023 mol-1 0.59 situation is undergoing international re­ Faraday constant, NAe F 96 485.309(29) C mol- 1 0.30 view and will be rectified in the near electron molar mass M(e), Me 5.485 799 03(13) 10- 7 kg/mol 0.023 future 19, 20). muon molar mass M(µ),M µ 1.134 289 13(17) 10- 4 kg/mol 0.15 proton molar mass M(p),Mp 1.007 276 470(12) 10- 3 kg/mol 0.012 Since the fine-structure constant a, Hartree energy, e2/4 πϵ0a0 = 2R∞hc Eh 4.359 7482(26) 10-18 J 0.60 which is proportional to e(e/h), has in eV, Eh/{e} 27.211 3961(81) eV 0.30 changed by only 0.37 ppm, the increase molar gas constant R 8.314 510(70) Jmol-1K-1 8.4 in 2e/h is strongly correlated to an ap­ Boltzmann constant, R/NA k 1.380 658(12) 10-23 J K- 1 8.5 proximately equal fractional decrease in first radiation constant, 2πhc2 c1 3.741 7749(22) 10-16 W m 2 0.60 e. If e2/h is almost unchanged and e second radiation constant, hc/k c2 0.014 387 69(12) m K 8.4 decreases, the fractional decrease in h Wien displacement law constant, b = λmaxT = c 2/4.965 114 23... b 2.897 756(24) 10-3 m K 8.4 must be twice as great. Furthermore, the Stefan-Boltzmann constant, a 5.670 51(19) 10-8W m -2 K- 4 34 quantity NAh is proportional to a2 ; a (π2/60)k4/h3c2 decrease in h is coupled with an increase in Na and with an increase (approxima­ tely half as large) in F. The changes from Table 3. Maintained units and standard values. the 1973 values of many quantities are A summary of ‘maintained’ units and ‘standard’ values and their relationship to SI units, based on a least-squares adjustment with 17 degrees of freedom. The digits in thus strongly correlated, and all of the parentheses are the one-standard-deviation uncertainty in the last digits of the given large changes can be directly linked to value. Since the uncertainties of many of these entries are correlated, the full covariance the change in Kv. This is seen in the matrix must be used in evaluating the uncertainties of quantities computed from them. comparison of the 1973 and 1986 re­ commended values of several constants Relative shown in Table 4. Quantity Symbol Value Units uncertainty (ppm) A major part of the difference bet­ ween 1973 and 1986 may be traced to the deletion, in 1973, of two Faraday electron volt, (e/C) J = {e} J eV 1.602 177 33(49) 10-19 J 0.30 (unified) atomic mass unit, u 1.6605402(10) 10- 27kg 0.59 determinations which seemed to be dis­ 1 u = mu = 1/12m(12C) crepant with the remaining data. In hind­ standard atmosphere atm 101325 Pa (exact) sight this 'discrepancy' was not that standard acceleration of gravity gn 9.80665 m s-2 (exact) severe. Adjustment No. 40 in that analy­ sis 13), which differs from the 1973 re­ ‘As-Maintained’ Electrical Units commended set (No. 41) only in its reten­ BIPM maintained ohm, Ω69- bi tion of the two Faraday determinations, ΩBI85 ≡ 69-BI (1 Jan 1985) Ω B I85 1 - 1.563(50)x10- 6 fi gives a value for 2e/h that is 5.3 ppm = 0.999998 437(50) fi 0.050 higher than the 1973 recommendation Drift rate of Ω69-BI -0.0566(15) µΩ/a and only 2.5 ± 2.0 ppm lower than the BIPM maintained volt, V76—BI 1 - 7.59(30)x10-6 V present value. It is important to reco­ V 76- b i = 483 594 GHz(h/2e) = 0.999 992 41(30) V 0.30 gnize, however, that there are no similar

BIPM maintained ampere, A BI85 1 — 6.03(30)x10-6 A data discrepancies in the present analy­ A BIPM = V76—BI/ Ω 69—BI = 0.999 993 97(30) A 0.30 sis ; the deleted data have been either ex­ tremely discrepant or of very low weight X-Ray Standards (or both). Thus, it is improbable that any Cu x-unit : xu(CuKα1) 1.002 077 89(70) 10-13 m 0.70 future reassessment of the current data λ(CuKα1) = 1537.400 xu could change the recommendations of Mo x-unit : xu(MoKα1) 1.002099 38(45) 10- 13 m 0.45 the present analysis by as much as two λ(MoKα1) = 707.831 xu standard deviations. Å* : A* 1.000 014 81(92) 10-10m 0.92 λA(WKα1) = 0.209 100 Å* The 1986 analysis does not separa­ lattice spacing of Si a 0.543 101 96(11) nm 0.21 tely consider those data that are inde­ (in vacuum, 22.5 °C), + pendent of quantum electrodynamics d220

a/88= (WQED data), as was done in 1969 and d220 192.015 540(40) pm 0.21 molar volume of Si, Vm(Si) 12.058 8179(89) cm3/ mol 0.74 1973. If the measurements that depend M(Si)/p(Si) = NAa3/ 8 on QED information (the electron ano­ malous moment and the muonium hy- + The lattice spacing of single-crystal Si can vary by parts in 107 depending on the preparation perfine-structure) are deleted, the re­ process. Measurements at PTB indicate also the possibility of distortions from exact cubic maining 20 items give a-1 = symmetry of the order of 0.2 ppm. 137.035 9846(94). This differs from the 67 Table 4. Comparison of 1973 and 1986 adjustments 11. Cohen E.R., Crowe K.M. and Du Mond fundamental constants", Natl. Bur. Stand. J.W.M., Fundamental Constants of Physics Report NBSIR 81-2426 (Jan. 1982). uncertainties change from (Interscience Publishers, New York) 1957. 16. Cohen E.R., "Extended' least squares". 1973 of recommended 12. Taylor B.N., Parker W.H. and Langenberg Report SCTR-76-1 (Rockwell International recommended values D.N., Rev. Mod. Phys. 41 (1969) 375; also Science Center) Jan. 1976. quantity value (ppm) published as The Fundamental Constants 17. Cohen E.R., "An extended least-squares (ppm) 1973 1986 and Quantum Electrodynamics (Academic algorithm for treating inconsistent data", α-1 -0.37 0.82 0.045 Press, New York). Report SCTR-78-11 (Rockwell International e -7.4 2.9 0.30 13. Cohen E.R. and Taylor B.N., J. Phys. Science Center) ; see also ref. 4, p. 391. h -15.2 5.4 0.60 Chem. Ref. Data 2 (1973) 663. 18. Cohen E.R., in "Metrology and Fun­ me -15.8 5.1 0.59 14. Tuninskii V.S. and Kholin S.V., "Concern­ damental Constants", Proceedings of the Na +15.2 5.1 0.59 ing changes in the methods for adjusting the International School of Physics 'Enrico Fer­ mp/m e +0.64 0.38 0.020 physical constants". Internal Report, Men­ mi', Course LXVIII, eds. A. Ferro-Milone, P. F +7.8 2.8 0.30 deleyev Research Institute of Metrology Giacomo and S. Leschiutta (North Holland, 2e/h +7.8 2.6 0.30 (VNIIM), Leningrad; Metrologiya 8 (1975) 3 Amsterdam) 1980, p. 581. [in Russian]. 19. Taylor B.N., J. Res. Natl. Bur. Stand. 91 recommended value by 0.036 ± 0.059 15. Taylor B.N., "Numerical comparisons of (1986) 299. ppm. The WQED value of KV differs from several algorithms for treating inconsistent 20. Taylor B.N., J. Res. Natl. Bur. Stand. 92 the recommended value by 0.01 ± 1.03 data in a least-squares adjustment of the (1987) 55. ppm. There is clearly no basis for any distinction between QED and WQED data. By deleting the quantum Hall effect International Cooperation Strengthens European Optics (QHE) data from the analysis one may in­ vestigate the validity of the theoretical Two events, of considerable importance to the future of optical technology in relation RH = h/e2. If the QHE data are Europe, marked the opening of the recent Fourth International Symposium on deleted, the value of a-1 becomes Optical and Optoelectronic Applied Science and Engineering, arranged by SPIE/ 137.0359884(79); the difference from ANRT at the Hague in the Netherlands. These were the creation of a new association the recommended value is -0.009 ± of West European Optical Societies called EUROPTICA, and the declaration of a 0. 071ppm. A value of α-1 from the Hall Memorandum of Understanding (MOU) between EUROPTICA, EPS (European Phy­ resistance data and the direct ohm de­ sical Society) and SPIE — The International Society for Optical Engineering based in terminations yields 137.0359943(127). the USA. This differs by only 0.043 ± 0.085 ppm In recent years, as the pace of developments in optical science and engineering from the value above. Thus, based on has increased, and more European countries have formed their own Optical Socie­ the presently available observational ties, and also as a result of the now established trend, led by SPIE, for the organiza­ data, there is no evidence of any discre­ tion of large multidisciplinary conferences, the need has arisen for the creation of a pancy in the QHE theory at the current focal point through which future major international events could be channelled. As levels of precision. a start, the partners of the MOU have agreed to collaborate in the organization of one REFERENCES major meeting in Europe each year to be known as the International Conference on 1. Comptes Rendus des Séances de la 17e Optical Science and Engineering. This title has been chosen to reflect the interests CGPM (BIPM, Sèvres, France) 1983. of EPS in optical science, those of EUROPTICA in the industrial application of new 2. Hudson R.P. (editor), Metrologia 19 developments in optics technology and the professional interests of optical engi­ (1984) 163. neers through their membership of SPIE. Plans are already well advanced for the first 3. Deslattes R.D. et al., Phys. Rev. Lett. 33 such meeting to be held from 19-23 September 1988 at the Hamburg Conference (1974) 463. Centre (CCH). The second meeting will be in Paris from 24-28 April 1989 at the 4. Seyfried P. et al., In Precision Measure­ Palais des Congrès de la Porte Maillot. ment and Fundamental Constants II, eds. This new partnership of non-profit bodies representing the interests of over B.N. Taylor and W.D. Phillips, Natl. Bur. Stand. (US), Spec. Publ. 617 (U.S. Govt. Prin­ 20,000 scientists, engineers and technologists in optics is expected to develop, in ting Office, Washington, DC) 1984, p. 313. the course of time, to encompass other services to the optical community in the 5. Van Dyck R.S. jr, Schwinberg P.B. and fields of education, exhibitions and publishing. The partnership will have the addi­ Dehmelt H.G., in Atomic Physics - 9, eds. tional benefit of speeding up the transfer of technology into industry and helping to R.S. van Dyck jr and E.N. Fortson (World promote international trade. These activities will be managed by an appointed Joint Scientific Publishing Co., Singapore) p. 38. Policy Committee (JPC) with the following representation : 6. Kinoshita T. and Sapirstein J., Op. Cit., p. 38. EPS — European Physical EUROPTICA including SPIE — The International So­ 7. Von Klitzing K., Dorda G. and Pepper M., Society Europtica Services ciety of Optical Engineering Phys. Rev. Lett. 45 (1980) 494. H.A. Ferweda, P. Bosec, ESSILOR L.R. Baker, Technical Director 8. Imry Y., 'Klaus von Klitzing', Europhys. Universiteit Groningen H. Walter, Director (Optics), Sira Ltd., News, 16 (1985) 11/12. G. Thomas, EPS, Research and Development, Chairman of JPC 9. Cohen E.R. and Taylor B.N., 'The 1986 Ad­ Secretary of JPC Rodenstock B.J. Thompson, Provost, justment of the Fundamental Physical Con­ H. Tiziani, Universität P. Zaleski, Director, ANRT University of Rochester stants', CODATA Bulletin No. 63 (Interna­ Stuttgart, Institut für W.L. Wolfe, tional Council of Scientific Unions — Com­ Technische Optik, University of Arizona, mittee on Data for Science and Technology Vice-chairman of JPC Optical Science Center (CODATA), 51, Blvd de Montmorency, 75016 Paris, France) Nov. 1986. 10. "Recommended Consistent Values of For further information contact: Dr. L.R. Baker the Fundamental Physical Constants, Sira Ltd., South Hill, Chislehurst 1973", CODATA Bulletin No. 11 (ICSU, Paris) Kent BR7 5EH, UK 1973. Tel.: (1) 467 26 36 Telex: 896 649

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